# Is there an analytical expression for the distribution of the max of a normal k sample?

For example:

k <- 100
R <- 10000
max.g <- numeric(R)

for(i in 1:R) max.g [i] <- max(rnorm(k))

hist(max.g)  # We can see it's right tailed...

I remember once encountering that there is a name for this type of distributions, but the name alludes me.

Properly normalized, it's closely approximated by a Gumbel distribution as shown by Extreme value theory. Alternative names are provided in the links.

• Thank you "Extreme value theory" was what I was looking for! – Tal Galili Oct 4 '10 at 18:59

You will find exact expressions for the full pdf of the $n^{th}$ order statistics (as a function of $n$, the sample size) in the following paper:

Percentage Points and Modes of Order Statistics from the Normal Distribution

Shanti S. Gupta Source: Ann. Math. Statist. Volume 32, Number 3 (1961), 888-893.

Also includes exact expression for the medians and means of $n^{th}$ order statistics as a function of $n$ (i could type a few here but the paper is un-gated). Some of these expressions are surprisingly simple.

H/T to John D Cook for the pointer.

• Thanks for the reference. It's always fun to see early work (even when it was published in one's own lifetime, LOL!). The results are simple because they combine two simple results: (1) we know the pdf of the Normal distribution (which allows us to give a name to its cdf, the Gaussian integral,which has no closed form evaluation). (2) the standard expressions for distributions of the order statistics of any pdf. Thus you do obtain "exact" expressions for the order statistics, but in they end they must be polynomials in the Gaussian integral. (Computing them in 1961 wasn't easy.) – whuber Oct 4 '10 at 20:18
• @whuber:> Yes. the link will probably be less useful than the Gumbel approximation (except perhaps for the <strike> min and max <\Strike> largest and smallest draw) but i cited the paper nonetheless because, as you said, it has this 'head in the cloud' cachet about it :) – user603 Oct 4 '10 at 20:59